Infinitely many homoclinic solutions for different classes of fourth-order differential equations
Commun. Korean Math. Soc. 2022 Vol. 37, No. 1, 137-161
Published online January 3, 2022
Printed January 31, 2022
Mohsen Timoumi
Faculty of Sciences of Monastir
Abstract : In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation $$u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in\mathbb{R} \leqno(1)$$ where $a(x)$ is not required to be either positive or coercive, and $F(x,u)=\int^{u}_{0}f(x,v)dv$ is of subquadratic or superquadratic growth as $\left|u\right|\rightarrow\infty$, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as $|u|\rightarrow\infty$). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.
Keywords : Fourth-order differential equations, homoclinic solutions, critical points, subquadratic growth, superquadratic growth, local conditions
MSC numbers : Primary 34C37, 58E05, 70H05
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